The $n\text{th}$ partial sum of the series $\sum\limits_{n=1}^{\infty }{{{a}_{n}}}$ is given by ${{S}_{n}}=\frac{n-1}{n+1}$. $\sum\limits_{n=1}^{6}{{{a}_{n}}}=$
Explanation: $\sum\limits_{n=1}^{6}{{{a}_{n}}}~~~$ is the same as $~~~{{S}_{6}}\,$. ${{S}_{6}}=\frac{6-1}{6+1}=\frac{5}{7}\,$.